Transcript Part_2b_-_Overview_o..

Randomized
Clinical Trials
(RCT)
Randomized
Clinical Trials
(RCT)
Designed to compare two or more treatment
groups for a statistically significant difference
between them – i.e., beyond random chance –
often measured via a “p-value” (e.g., p < .05).
Examples: Drug vs. Placebo, Drugs vs. Surgery, New Tx vs. Standard Tx
 Let X =  cholesterol level (mg/dL); possible expected distributions:
Experiment
Treatment
population
Control
population
significant?
1
2
H 0 : 1  2
X
0
Patients
satisfying
inclusion
criteria
R
A
N
D
O
M
I
Z
E
Treatment
Arm
End of Study
RANDOM
SAMPLES
Control
Arm
T-test
F-test
(ANOVA)
Randomized
Clinical Trials
(RCT)
Designed to compare two or more treatment
groups for a statistically significant difference
between them – i.e., beyond random chance –
often measured via a “p-value” (e.g., p < .05).
Examples: Drug vs. Placebo, Drugs vs. Surgery, New Tx vs. Standard Tx
 Let X =  cholesterol level (mg/dL) from baseline, on same patients
Experiment
Post-Tx
population
Pre-Tx
population
significant?
1
2
H 0 : 1  2
X
0
Patients
satisfying
inclusion
criteria
Pre-Tx
Arm
End of Study
PAIRED
SAMPLES
Post-Tx
Arm
Paired T-test,
ANOVA F-test
“repeated
measures”
Randomized
Clinical Trials
(RCT)
Designed to compare two or more treatment
groups for a statistically significant difference
between them – i.e., beyond random chance –
often measured via a “p-value” (e.g., p < .05).
Examples: Drug vs. Placebo, Drugs vs. Surgery, New Tx vs. Standard Tx
 Let T = Survival time (months); population survival curves:
S(t) = P(T > t)
1
survival
probability
Kaplan-Meier
estimates
AUC difference
significant?
S2(t)
Control
S1(t)
Treatment
0
H 0 : S1 (t )  S2 (t )
T
End of Study
Log-Rank Test,
Cox Proportional
Hazards Model
Case-Control
studies
Cohort
studies
Observational study designs that test for a statistically significant association
between a disease D and exposure E to a potential risk (or protective) factor,
measured via “odds ratio,” “relative risk,” etc. Lung cancer / Smoking
Case-Control
studies
Cohort
studies
PRESENT
PAST
FUTURE
cases
E+ vs. E– ?
controls
reference group
D+ vs. D– E+ vs. E–
 relatively easy and inexpensive
subject to faulty records, “recall bias”
D+ vs. D– ?
 measures direct effect of E on D
expensive, extremely lengthy…
Example: Framingham, MA study
Both types of study yield a 22 “contingency table” for binary variables D and E:
D+
D–
E+
a
b
a+b
E–
c
d
c+d
a+c b+d
n
where a, b, c, d are the
observed counts of
individuals in each cell.
H0: No association
between D and E.
End of Study
Chi-squared Test
McNemar Test
(for
paired
casecontrol study designs)
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Y
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
Correlation
Coefficient
JAMA. 2003;290:1486-1493
measures the strength
of linear association
between X and Y
Scatterplot
–1
0
negative linear
correlation
X
+1
positive linear
correlation
r
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Y
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
Correlation
Coefficient
JAMA. 2003;290:1486-1493
measures the strength
of linear association
between X and Y
Scatterplot
–1
0
negative linear
correlation
X
+1
positive linear
correlation
r
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Y
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
Correlation
Coefficient
JAMA. 2003;290:1486-1493
measures the strength
of linear association
between X and Y
Scatterplot
–1
0
negative linear
correlation
X
+1
positive linear
correlation
r
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
Correlation
Coefficient
measures the strength
of linear association
between X and Y
For this example, r = –0.387
(weak, negative linear correl)
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
residuals
Simple Linear Regression
gives the “best” line
that fits the data.
Regression
Methods
Want the unique line that minimizes
the sum of the squared residuals.
For this example, r = –0.387
(weak, negative linear correl)
As seen, testing for association between categorical variables – such as
disease D and exposure E – can generally be done via a Chi-squared Test.
But what if the two variables – say, X and Y – are numerical measurements?
Furthermore, if sample data does suggest that one exists, what is the nature
of that association, and how can it be quantified, or modeled via Y = f (X)?
residuals
Simple Linear Regression
gives the “least squares”
regression line.
Regression
Methods
Want the unique line that minimizes
the sum of the squared residuals.
For this example, r = –0.387
(weak, negative linear correl)
Y = 8.790 – 4.733 X (p = .0055)
It can also be shown that the proportion of
total variability in the data that is accounted for
by the line is equal to r 2, which in this case,
= (–0.387)2 = 0.1497 (15%)... very small.
• Polynomial Regression – predictors X, X2, X3,…
• Multilinear Regression – independent predictors X1, X2,…
w/o or w/ interaction (e.g., X5 X8)
• Logistic Regression – binary response Y (= 0 or 1)
• Transformations of data, e.g., semi-log, log-log,…
• Generalized Linear Models
• Nonlinear Models
• many more…
Numerical (Quantitative)
e.g., $ Annual Income
2 POPULATIONS:
Independent
σ1
e.g., RCT
Sample 1
X
n1 , x1 , s1
No
H0: 1 = 2
Yes
Paired (Matched)
σ2
Normally distributed?
Sample 2
e.g., Pre- vs. Post-
n2 , x2 , s2
Yes
No
• Q-Q plots
Yes
n1 , n2  30?
No
• Shapiro-Wilk
• Anderson-Darling
• others…
Yes
No
Equivariance?
“Nonparametric
Tests”
Wilcoxon
Rank Sum
(aka MannWhitney U)
• F-test
• Bartlett
• others… “Approximate” T
2-sample T
(w/o pooling)
2-sample T
(w/ pooling)
• Satterwaithe
• Welch
 2 POPULATIONS:
KruskalWallis
• ANOVA F-test
• Regression Methods
Various
modifications
“Nonparametric
Tests”
Paired T
ANOVA F-test
(w/ “repeated
measures”
or “blocking”)
• Sign Test
• Wilcoxon
Signed Rank
• Friedman
• Kendall’s W
• others…
Categorical (Qualitative)
e.g., Income Level: Low, Mid, High
 2 CATEGORIES per
each of two variables:
H0: “There is no association between
(the categories of) I and
(the categories of) J.”
r × c contingency table
J
1
I
2
3
Chi-squared Tests
•••
c
 Test of Independence
(1 population, 2 categorical variables)
1
 Test of Homogeneity
(2 populations, 1 categorical variable)
2
 “Goodness-of-Fit” Test
(1 population, 1 categorical variable)
3
•••
r
etc.
 Modifications
• McNemar Test for paired
2 × 2 categorical data, to control
for “confounding variables”
e.g., case-control studies
• Fisher’s Exact Test for small
“expected values” (< 5) to avoid
possible “spurious significance”
Introduction to Basic Statistical Methods
Part 1: Statistics in a Nutshell
Part 2: Overview of Biostatistics:
“Which Test Do I Use??”
Sincere thanks to…
UWHC Scholarly Forum
May 21, 2014
• Judith Payne
Ismor Fischer, Ph.D.
UW Dept of Statistics
[email protected]
• Samantha Goodrich
• Heidi Miller
• Troy Lawrence
• YOU!
All slides posted at
http://www.stat.wisc.edu/~ifischer/Intro_Stat/UWHC